Przemysław Litewka

The exact thick arch finite element

Abstract The exact stiffness matrix is derived for a curved beam element with constant curvature. The plane two-node six degree-of-freedom element is considered in which effects of flexural, axial and shear deformations are taken into account. The analytical shape functions describing radial and tangential displacements as well as cross-section rotations are found in the algebraic–trigonometric form. They contain the coupled influences of shear and membrane effects. Based on these shape functions, using the strain energy formula, the stiffness matrix for shear flexible and compressible arch element is formulated. Obviously, this element is completely free of shear and membrane locking effects. The advantage of the elaborated element is its applicability to any combination of geometrical properties of the arch structure, e.g. the depth–length ratio of element. In presented numerical examples the shear and membrane influences on the displacements for various cases of boundary conditions and loading are investigated. The results coincide exactly with the analytical ones obtained for continuous arches.

AN EFFICIENT CURVED BEAM FINITE ELEMENT

The plane two-node curved beam finite element with six degrees of freedom is considered. Knowing the set of 18 exact shape functions their approximation is derived using the expansion of the trigonometric functions in the power series. Unlike the ones commonly used in the FEM analysis the functions suggested by the authors have the coefficients dependent on the geometrical and physical properties of the element. From the strain energy formula the stiffness matrix of the element is determined. It is very simple and can be split into components responsible for bending, shear and axial forces influences on the displacements. The proposed element is totally free of the shear and membrane locking effects. It can be referred to the shear-flexible (parameter d) and compressible (parameter e) systems. Neglecting d or e yields the finite elements in all necessary combinations, i.e. curved Euler–Bernoulli beam or curved Timoshenko beam with or without the membrane effect. Applying the elaborated element in the calculations a very good convergence to the analytical results can be obtained even with a very coarse mesh without the commonly adopted corrections as reduced or selective integration or introduction of the stabilization matrices, additional constraints, etc., for the small depth–length ratio.

Smooth Frictional Contact between Beams in 3D

In this paper a smoothing procedure for the 3D beam-to-beam contact is presented. A smooth segment is based on current position vectors of three nodes for two adjacent finite elements. The approximated fragment of a 3D curve modeling a beam axis spans between the centre points of these elements. The curve is described parametrically using three Hermite polynomials. The four boundary conditions used to determine the coefficients for each of these polynomials involve co-ordinates and slopes at the curve ends. The slopes are defined in terms of the element nodal co-ordinates, too, so there is no dependence on nodal rotations and this formulation can be embedded in a beam analysis using any type of beam finite element. This geometric representation of the curve is incorporated into the 3D beam-to-beam frictional contact model with the penalty method used to enforce contact constraints. The residual vector and the corresponding tangent stiffness matrix are determined for the normal part of contact and for the stick or slip state of friction. A numerical example is presented to show the performance of the suggested smoothing procedure in a case of large frictional sliding.

Influence of elastic supports on non-linear steady-state vibrations of Zener material plates

The paper reports numerical results of analyses of steady-state harmonic vibrations of von Karman non-linear plates made from Zener material with various elastic support conditions. Influences of s...

Thermo-Mechanical Coupling in Beam-to-Beam Contact

Contact including coupling between mechanical and thermal fields constitutes a complicated problem because the mutual influences between displacements or strains and temperature are manifested in many different ways. A detailed description of various issues related to the thermo-mechanical coupling in contact can be found in the monograph by [6]. In this paper a formulation of a beam-to-beam contact element for the coupled thermo-mechanical field is presented. Contact surfaces are considered ideally smooth, so that the heat transfer due to radiation and convection in the micro cavities is neglected. In the physi-cal model of the beam material, only the linear thermal expansion is included and all the physical parameters are treated as independent of time and temperature.

The Beam-to-Beam Contact Smoothing with Bezier’s Curves and Hermite’s Polynomials

This contribution concerns a key issue in numerical treatment of contact by the FEM, i.e. ensuring the C1-continuity of contacting facets. After a discussion of basic issues of the frictional beam-to-beam contact two methods of curve construction – inscribed curve and node-preserving ones, are presented. Each of them is combined with a curve representation by Hermite’s polynomials and Bezier’s curves. In this way four possibilities of determination of a position vector for an arbitrary point on the curve are obtained. The resulting four different beam contact finite elements are briefly described. Then, on a base of this derivation and results of representative numerical examples, the elements are compared. To this end several criteria as: accuracy of curve approximation, number of degrees of freedom, code length, computer time and smoothing effectiveness, are used. The final conclusion is that despite the complicated, long code the inscribed curve method combined with the Hermite’s polynomials should be preferred.

Thermo-mechanical Coupling

Analysis of contact with inclusion of coupling between mechanical and thermal fields is a complicated problem because the mutual influences between displacements or strains and temperature are manifested in many different ways. The aspects involved include: heat flow through a real contact area resulting from roughness of contacting surfaces, heat flow through a gas between the bodies, heat flow through radiation; frictional heating; dependence of material properties like elasticity moduli, friction coefficient or heat conduction coefficient on temperature, etc. The more detailed description of various issues related to the thermo-mechanical coupling in contact can be found in the monographs by Wriggers (2002) and Laursen (2002). One can find there numerous references to the papers and other monographs devoted to the problem of the heat conduction in contact. This phenomenon requires a precise description of geometry of bodies surfaces in the micro scale and a development of a thermo-mechanical physical law for the contact. To this end statistical methods can be used (Cooper et al. 1969 and Song and Yovanowic 1987). The numerical solution to this problem was a subject of the papers by Zavarise (1991), Zavarise et al. (1992) as well as Wriggers and Zavarise (1993b). Another problem is related to the frictional heating due to the contact. This topic was considered, for instance, in the papers by Wriggers and Miehe (1992) or by Zavarise et al. (1995, 2005).

Thermo-electro-mechanical coupling in beam-to-beam contact

This paper is a further step towards the full analysis of the complex behaviour of contacting beams in the coupled thermo-electro-mechanical field. Now we add to our previous work the coupling with the thermal field. The coupling between electric, mechanical and thermal fields is manifested in: dependence of material parameters on the temperature, frictional heating, heat generation by the electric current flow, dependence of the contact area on the temperature, dependence of thermal and electric fields on the change of the contact area and the contact point position. In the present preliminary analysis we take into account only the last aspect. We also consider the indirect influence of thermal contact on the mechanical field using the thermo-mechanical beam element. The contact constraints are enforced with the penalty method within the finite element technique. It is assumed that the heat flow in the contact area is unlimited which leads to the equalling of temperatures between two contacting bodies. This constraint is also introduced by the penalty method. The set of governing equations including the coupling is solved by the monolithic scheme. The problem is nonlinear, hence the linearized version of equations is derived. The consistent linearization leading to the consistent tangent stiffness matrix and the corresponding residual vector is performed to apply efficiently the Newton-Raphson method and to ensure the quadratic convergence. Some numerical examples are presented to show the efficiency of the suggested approach.

3D beam-to-beam contact within coupled electromechanical fields: a finite element model

In this paper a 3D beam-to-beam contact element is presented, to deal with contact problems in the coupled electric mechanical fields. The beams are supposed to get in contact in a pointwise manner, the detection of the contact points and the computation of all contributions are carried out using a fully symmetric treatment of the two beams. Concerning the mechanical field, Hertz theory of contact for elastic bodies is considered. The contact area is varying according to the beamto-beam angle, being circular only in the case of perpendicular beams. This variation of the shape is taken into account too. The problem is semi-coupled: the mechanical field influences the electric one because of the dependence of the voltage distribution on the contact area. Within the finite element discretization, the mechanical and the electric treatment of the beam element is formulated in the usual way, considering nodal displacements and voltages as main unknowns. The electromechanical contact constraints are enforced with the penalty method. Starting from the virtual work equation the consistent linearization of all contributions is computed to achieve the quadratic convergence within the Newton-Raphson iterative scheme. The complete set of equations arranged in a matrix form suitable for the finite element implementation is solved with a monolithic approach. Finally some numerical examples are discussed to show the effectiveness of the model. 104 D.P. Boso, P. Litewka, B.A. Schrefler, and P. Wriggers